Fractal concepts were first introduced for use in antenna array theory by Kim and Jaggard. See, Y. Kim et al., “The Fractal Random Array,” Proc. IEEE, Vol. 74, No. 9, pp. 1278–1280, 1986. A design methodology was developed for quasi-random arrays based on properties of random fractals. In other words, random fractals were used to generate array configurations that are somewhere between completely ordered (i.e., periodic) and completely disordered (i.e., random). The main advantage of this technique is that it yields sparse arrays that possess relatively low sidelobes (a feature typically associated with periodic arrays but not random arrays) which are also robust (a feature typically associated with random arrays but not periodic arrays). More recently, the fact that deterministic fractal arrays can be generated recursively (i.e., via successive stages of growth starting from a simple generating array) has been exploited to develop rapid algorithms for use in efficient radiation pattern computations and adaptive beamforming, especially for arrays with multiple stages of growth that contain a relatively large number of elements. See, D. H. Werner et. al., “Fractal Antenna Engineering: The Theory and Design of Fractal Antenna Arrays,” IEEE Antennas and Propagation Magazine, Vol. 41, No. 5, pp. 37–59, October 1999. It was also demonstrated that fractal arrays generated in this recursive fashion are examples of deterministically thinned arrays. A more comprehensive overview of these and other topics related to the theory and design of fractal arrays may be found in D. H. Werner and R. Mittra, Frontiers in Electromagnetics (IEEE Press, 2000).
Techniques based on simulated annealing and genetic algorithms have been investigated for optimization of thinned arrays. See, D. J. O'Neill, “Element Placement in Thinned Arrays Using Genetic Algorithms,” OCEANS '94, Oceans Engineering for Today's Technology and Tomorrows Preservation, Conference Proceedings, Vol. 2, pp. 301–306, 199; G. P. Junker et al., “Genetic Algorithm Optimization of Antenna Arrays with Variable Interelement Spacings,” 1998 IEEE Antennas and Propagation Society International Symposium, AP-S Digest, Vol. 1, pp. 50–53, 1998; C. A. Meijer, “Simulated Annealing in the Design of Thinned Arrays Having Low Sidelobe Levels,” COMSIG'98, Proceedings of the 1998 South African Symposium on Communications and Signal Processing, pp. 361–366, 1998; A. Trucco et al., “Stochastic Optimization of Linear Sparse Arrays,” IEEE Journal of Oceanic Engineering, Vol. 24, No. 3, pp. 291–299, July 1999; R. L. Haupt, “Thinned Arrays Using Genetic Algorithms,” IEEE Trans. Antennas Propagat., Vol. 42, No. 7, pp. 993–999, July 1994. A typical scenario involves optimizing an array configuration to yield the lowest possible side lobe levels by starting with a fully populated uniformly spaced array and either removing certain elements or perturbing the existing element locations. Genetic algorithm techniques have been developed for evolving thinned aperiodic phased arrays with reduced grating lobes when steered over large scan angles. See, M. G. Bray et al., “Thinned Aperiodic Linear Phased Array Optimization for Reduced Grating Lobes During Scanning with Input Impedance Bounds, “Proceedings of the 2001 IEEE Antennas and Propagation Society International Symposium, Boston, Mass., Vol. 3, pp. 688–691, July 2001; M. G. Bray et al.,” Matching Network Design Using Genetic Algorithms for Impedance Constrained Thinned Arrays,” Proceedings of the 2002 IEEE Antennas and Propagation Society International Symposium, San Antonio, Tex., Vol. 1, pp. 528–531, June 2001; M. G. Bray et al., “Optimization of Thinned Aperiodic Linear Phased Arrays Using Genetic Algorithms to Reduce Grating Lobes During Scanning,” IEEE Transactions on Antennas and Propagation, Vol. 50, No. 12, pp. 1732–1742, December 2002. The optimization procedures have proven to be extremely versatile and robust design tools. However, one of the main drawbacks in these cases is that the design process is not based on simple deterministic design rules and leads to arrays with non-uniformly spaced elements.